The ridges on the base of the oil screen seal it. Check closely for damage. That said, that thing doesn't just start leaking. Leakage behind the metal bar plate is often mis-diagnosed as that screen leaking. The fit where the oil hose enters the tank is a very common spot to leak. The pressure from the bar plate creates the seal where the hose enters the tank. As the saw ages that plate relaxes and a leak develops. You may be able to re-bend the plate or add some sealant in that area if you discover that as the actual source of your leak.
Thanks for the suggestions.
I know it at least leaks some from this point. I would suspect it is at least 75%. I have not completely torn the saw apart yet (I am going to be replacing the intake clamp, boot and partition soon as the "bellows" comes in), but so far I have not seen any other way for oil to get out of the tank but through this "screen". I do not see any "oil hose" except where it comes off of the pump and up to where it oils the bar.
I have a Jonsered CS2255 that was leaking way worse than this Husqvarna 350 (if you can believe that) and I used a product called "Seal-All" on the rubber hose where it pushes into the tank. So far, so good. No more leaks on that saw. I think I am going to go with 1184 (Three Bond) to seal this "screen" on this saw, but first I am going to finish the tear down and clean up.
Again, thanks for your input on this matter.